So I had a look at
E. Borel: Leçons sur les séries divergentes (1901)
In chapter V, page 156 he explains the Mittag-Leffler expansion.
Mostly I fighted my way through the text with an online translator.
I found the following interesting formulas, where
is the function we search the expansion of:
=\sum_{\lambda_1=0}^{n^{2n}}\sum_{\lambda_2=0}^{n^{2n-2}}<br />
\dots\sum_{\lambda_n=0}^{n^2} \frac{\phi^{(\lambda_1+\dots+\lambda_n)}(0)}{\lambda_1!\dots\lambda_n!} \left(\frac{x}{n}\right)^{\lambda_1+\dots+\lambda_n}<br />
)
=g_0(x)=\phi(0))
=g_n(x)-g_{n-1}(x))
for 
,
=\sum_{n=0}^\infty G_n(x))
E. Borel: Leçons sur les séries divergentes (1901)
In chapter V, page 156 he explains the Mittag-Leffler expansion.
Mostly I fighted my way through the text with an online translator.
I found the following interesting formulas, where
